climxrrelem - Mathbox for Glauco Siliprandi

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Theorem climxrrelem 46577

Description: If a sequence ranging over the extended reals converges w.r.t. the standard topology on the complex numbers, then there exists an upper set of the integers over which the function is real-valued. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

**Hypotheses**
Ref	Expression
climxrrelem.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
climxrrelem.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
climxrrelem.f	⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
climxrrelem.c	⊢ (𝜑 → 𝐹 ⇝ 𝐴)
climxrrelem.d	⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
climxrrelem.p	⊢ ((𝜑 ∧ +∞ ∈ ℂ) → 𝐷 ≤ (abs‘(+∞ − 𝐴)))
climxrrelem.n	⊢ ((𝜑 ∧ -∞ ∈ ℂ) → 𝐷 ≤ (abs‘(-∞ − 𝐴)))

**Assertion**
Ref	Expression
climxrrelem	⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ)

Distinct variable groups: 𝐴,𝑗 𝐷,𝑗 𝑗,𝐹 𝑗,𝑀 𝑗,𝑍 𝜑,𝑗

Proof of Theorem climxrrelem Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1947	. . . . 5 ⊢ Ⅎ𝑘𝜑
2		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
3		nfra1 3286	. . . . . 6 ⊢ Ⅎ𝑘∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)
4	2, 3	nfan 1932	. . . . 5 ⊢ Ⅎ𝑘(𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
5	1, 4	nfan 1932	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)))
6		climxrrelem.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ_≥‘𝑀)
7	6	uztrn2 12906	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
8	7	adantll 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
9		climxrrelem.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
10	9	fdmd 6713	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝑍)
11	10	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → dom 𝐹 = 𝑍)
12	8, 11	eleqtrrd 2863	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
13	12	adantlrr 734	. . . . 5 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
14		simpll 779	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝜑)
15	8	adantlrr 734	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
16		rspa 3251	. . . . . . . 8 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
17	16	adantll 727	. . . . . . 7 ⊢ (((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
18	17	adantll 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
19	9	ffvelcdmda 7077	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ^*)
20	19	3adant3 1150	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → (𝐹‘𝑘) ∈ ℝ^*)
21		simpll 779	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → 𝜑)
22		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = -∞) → (𝐹‘𝑘) = -∞)
23		simpl 488	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = -∞) → (𝐹‘𝑘) ∈ ℂ)
24	22, 23	eqeltrrd 2861	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = -∞) → -∞ ∈ ℂ)
25	24	adantll 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → -∞ ∈ ℂ)
26		climxrrelem.n	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → 𝐷 ≤ (abs‘(-∞ − 𝐴)))
27	21, 25, 26	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → 𝐷 ≤ (abs‘(-∞ − 𝐴)))
28	27	adantlrr 734	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → 𝐷 ≤ (abs‘(-∞ − 𝐴)))
29		fvoveq1 7436	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) = -∞ → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(-∞ − 𝐴)))
30	29	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = -∞) → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(-∞ − 𝐴)))
31		simpl 488	. . . . . . . . . . . . . 14 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = -∞) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)
32	30, 31	eqbrtrrd 5129	. . . . . . . . . . . . 13 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞ − 𝐴)) < 𝐷)
33	32	adantll 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞ − 𝐴)) < 𝐷)
34	33	adantlrl 733	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞ − 𝐴)) < 𝐷)
35		climxrrelem.c	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
36	6	fvexi 6892	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑍 ∈ V
37	36	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑍 ∈ V)
38	9, 37	fexd 7226	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹 ∈ V)
39		eqidd 2761	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = (𝐹‘𝑘))
40	38, 39	clim 15581	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))))
41	35, 40	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))
42	41	simpld 500	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℂ)
43	42	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → 𝐴 ∈ ℂ)
44	25, 43	subcld 11593	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → (-∞ − 𝐴) ∈ ℂ)
45	44	abscld 15526	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞ − 𝐴)) ∈ ℝ)
46	45	adantlrr 734	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞ − 𝐴)) ∈ ℝ)
47		climxrrelem.d	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
48	47	rpred 13086	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ ℝ)
49	48	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → 𝐷 ∈ ℝ)
50	46, 49	ltnled 11381	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → ((abs‘(-∞ − 𝐴)) < 𝐷 ↔ ¬ 𝐷 ≤ (abs‘(-∞ − 𝐴))))
51	34, 50	mpbid 235	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → ¬ 𝐷 ≤ (abs‘(-∞ − 𝐴)))
52	28, 51	pm2.65da 829	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → ¬ (𝐹‘𝑘) = -∞)
53	52	3adant2 1149	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → ¬ (𝐹‘𝑘) = -∞)
54	53	neqned 2962	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → (𝐹‘𝑘) ≠ -∞)
55		simpll 779	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → 𝜑)
56		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = +∞) → (𝐹‘𝑘) = +∞)
57		simpl 488	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = +∞) → (𝐹‘𝑘) ∈ ℂ)
58	56, 57	eqeltrrd 2861	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = +∞) → +∞ ∈ ℂ)
59	58	adantll 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → +∞ ∈ ℂ)
60		climxrrelem.p	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → 𝐷 ≤ (abs‘(+∞ − 𝐴)))
61	55, 59, 60	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → 𝐷 ≤ (abs‘(+∞ − 𝐴)))
62	61	adantlrr 734	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → 𝐷 ≤ (abs‘(+∞ − 𝐴)))
63		fvoveq1 7436	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) = +∞ → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(+∞ − 𝐴)))
64	63	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = +∞) → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(+∞ − 𝐴)))
65		simpl 488	. . . . . . . . . . . . . 14 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = +∞) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)
66	64, 65	eqbrtrrd 5129	. . . . . . . . . . . . 13 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞ − 𝐴)) < 𝐷)
67	66	adantll 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞ − 𝐴)) < 𝐷)
68	67	adantlrl 733	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞ − 𝐴)) < 𝐷)
69	42	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → 𝐴 ∈ ℂ)
70	59, 69	subcld 11593	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → (+∞ − 𝐴) ∈ ℂ)
71	70	abscld 15526	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞ − 𝐴)) ∈ ℝ)
72	71	adantlrr 734	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞ − 𝐴)) ∈ ℝ)
73	48	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → 𝐷 ∈ ℝ)
74	72, 73	ltnled 11381	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → ((abs‘(+∞ − 𝐴)) < 𝐷 ↔ ¬ 𝐷 ≤ (abs‘(+∞ − 𝐴))))
75	68, 74	mpbid 235	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → ¬ 𝐷 ≤ (abs‘(+∞ − 𝐴)))
76	62, 75	pm2.65da 829	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → ¬ (𝐹‘𝑘) = +∞)
77	76	3adant2 1149	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → ¬ (𝐹‘𝑘) = +∞)
78	77	neqned 2962	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → (𝐹‘𝑘) ≠ +∞)
79	20, 54, 78	xrred 46194	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → (𝐹‘𝑘) ∈ ℝ)
80	14, 15, 18, 79	syl3anc 1398	. . . . 5 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ)
81	13, 80	jca 521	. . . 4 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ))
82	5, 81	ralrimia 3261	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ))
83	9	ffund 6707	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
84		ffvresb 7119	. . . . 5 ⊢ (Fun 𝐹 → ((𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ)))
85	83, 84	syl 18	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ)))
86	85	adantr 486	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) → ((𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ)))
87	82, 86	mpbird 260	. 2 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) → (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ)
88		breq2 5107	. . . . . 6 ⊢ (𝑥 = 𝐷 → ((abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
89	88	anbi2d 642	. . . . 5 ⊢ (𝑥 = 𝐷 → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)))
90	89	rexralbidv 3228	. . . 4 ⊢ (𝑥 = 𝐷 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)))
91	41	simprd 501	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
92	90, 91, 47	rspcdva 3577	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
93		climxrrelem.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
94	6	rexuz3 15436	. . . 4 ⊢ (𝑀 ∈ ℤ → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)))
95	93, 94	syl 18	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)))
96	92, 95	mpbird 260	. 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
97	87, 96	reximddv 3178	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ∃wrex 3086 Vcvv 3450 class class class wbr 5103 dom cdm 5655 ↾ cres 5657 Fun wfun 6527 ⟶wf 6529 ‘cfv 6533 (class class class)co 7413 ℂcc 11122 ℝcr 11123 +∞cpnf 11264 -∞cmnf 11265 ℝ^*cxr 11266 < clt 11267 ≤ cle 11268 − cmin 11465 ℤcz 12615 ℤ_≥cuz 12887 ℝ⁺crp 13042 abscabs 15321 ⇝ cli 15571

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-sup 9412 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-n0 12529 df-z 12616 df-uz 12888 df-rp 13043 df-seq 14066 df-exp 14126 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-clim 15575

This theorem is used by: climxrre 46578

W3C validator